Here is the full list of all 18 Aikagura achievements.
Defeated the Single Eye.
Defeated the headless samurai.
Defeated Syugenji Renin.
Defeated the red dragon.
Defeated the ammonite.
Defeated the hearse.
Defeated Yasaka Jinsuke.
Defeated Kushimaro.
Defeated the district guardian ninja master.
Defeated the white spider queen.
Defeated Okojo-no-mae.
Defeated Shokain Kochou.
Raised Strengthen Attack to level 5.
Raised Strengthen Endurance to level 5.
Raised Divine Arts to level 5.
Gained all sword skills.
Gained all spear skills.
Gained all divine art skills.